On counting and addition
When mathematical truths and their applicability to the physical world are discussed, there’s a certain kind of flawed (in my opinion) thinking that is often employed, and it comes out in sentences like “rocks behave isomorphically to integers, while clouds don’t”, or “two apples plus two apples is four apples, unless someone stole one from the bowl”, and so on. I’ll try to explain why I consider such reasoning flawed, and what are the more suitable descriptions of what’s going on with the apples and the rocks and the clouds and the drops of water and such.
Two mistakes combine together and create a shared state of confusion; the mistakes themselves are almost independent and I think they stand or fall separately.
1. The first mistake is conflating counting things and bringing things together spatially. One usually goes with the other because that’s how we learn to count, by looking at groups of things located closely together—but it doesn’t have to be this way. When we say “take 2 apples and add 2 more apples, now count—you have 4 apples”, we automatically imagine the two additional apples being brought in and placed next to the two original ones, but that’s just a mental crutch you can easily do without. Let me show you: suppose I ask you to consider two sheep in England and two more sheep in Australia—how many sheep are there together? You see the English sheep in your mind’s eye, there they are standing, and you count one, two. Now please resist the temptation to teleport the two Australian sheep and place them next in sequence. Instead, just fly with your mind’s eye all the way to Australia in a split second and home on that field—there they are—and continue counting: three, four. There, you just counted 2+2 sheep without bringing them together in space, in real life or your imagination. There’s nothing to it and you do it all the time—if someone asks you to count the number of chairs in a large and busy-looking living room, you don’t bring them together in your mind, you just gaze-travel over the room and count them off one by one.
Now consider the implication of that to clouds or other such objects. Suppose someone tells you “well, apples obey the law of arithmetics, but one drop of water plus another drop of water equals one larger drop of water, not two”—it should be clear that they are naively conflating spatial movement and adding/counting. Adding two apples and two apples is not a spatial operation of bringing them together, it’s a mental act of viewing them as one whole collection that can be counted. It’s just that bringing them spatially together, in reality or your imagimation, is the easiest way to carry that “viewing” out. For drops of water or clouds, the spatial operation becomes a distraction, so just resist it. One cloud plus one cloud most certainly equals two clouds: just count them in your mind, here’s one and there’s two, maybe drifting next to each other, or maybe they’re on separate continents. You don’t have to merge them—nothing about “+” says you do.
2. The second mistake is a sort of a map-territory confusion where we naively grant the territory the power of holding discrete objects for our needs. It may be helpful to realize and to remember that at least on our macro scale of reality, on the scale of things we perceive with our senses, discrete, separate objects are a feature of the map, not the territory; they exist in your mind, not the reality. In the reality, there’s just a lot of atoms everywhere (I’m simplifying; to be more thorough, think of elementary particles and many-worlds if you feel like it, and the virtual vacuum particles and so on) with no “natural” way of separating them into different conglomerates.
Normally, this really isn’t an important point to insist on, and there’s no harm whatsoever in just thinking of reality as being made up of objects like apples and clouds and whatnot. But imagine now an argument over an issue like “Was 2+2=4 before humans were around to invent that equation? I believe that long before humans, on an Earth devoid of life, when two rocks rolled onto a beach with three rocks already on it, there were five rocks altogether on the beach”. What’s really happening in that story? Well, there’s the spatial confusion dealt with above, when we find it easiest to imagine the two rocks to be added rolling into the scene. But even before that, zoom in on those three rocks on the beach. What are they? What business do you have saying there’s a “rock” there? There’s a bunch of atoms of various kinds, and lots of other atoms (of air, sand, etc.) around and next to those, with somewhat different densities and behavior, but no definite boundary between any of them. On the nanoscale, there’s constant exchange of atoms everywhere. To say “this is a discrete object, a rock” you need to be able to somehow ignore this inherently fuzzy boundary, maybe by picking a scale and smoothing out all the fine details below it—in human experience, the crudeness of our senses does that job for us, but our senses aren’t there in that picture. The reality doesn’t know or care about “rocks”—there’s just atoms.
This isn’t to say, I find it important to note, that the existence of three rocks on that beach is somehow a “subjective” claim. Imagine that there are alien nanobots everywhere on that Earth, recording everything faithfully on the nanoscale and storing the data somewhere for someone to discover; a billion years later, you come along, parse the data, reconstruct the scene—and of course you’ll then recognize that there were three rocks on the beach. The fact that there are three rocks on that beach is an objective fact of nature; it’s just that the meaning of that statement relies on the procedure of discretization being carried out, on someone or something defining what we consider a single discrete object and how we isolate it; and nature won’t do that for you. You do that with your brain. That’s why counting—and addition—are inherently mental acts carried out on mental constructs; on the map, not the territory.
All this is not to say that math is “invented” rather than “discovered”; I think my analysis is silent on that, and both platonism and formalism remain possible. It’s just that an example of five rocks on the beach before humans are around is not helpful in resolving that question—without human-style discretization you can’t meaningfully say it was five rocks there, rather than just a bunch of atoms. It may still be true—I certainly believe so—that the laws that govern the behavior of discrete objects, though they are normally mental entities, are universal and 2+2 was 4 before humans were around and in any alien mentality.
(It remains to acknowledge that reality may be discrete on the micro level—spacetime itself may be discrete, we can certainly speak of single photons, etc. This is irrelevant on the level of our everyday perceptions, however—the level of apples, rocks, clouds and so on).
Clear writing, clear thinking, much appreciated.
An aside: the Venus fly trap plant has fibers in its hinged petals. Touching one will not close it. Touching more than one with a delay between touches will not close it. Touching one or more fibers in succession will close it. This plant moves as if it can count and is aware of time. Learning that caused me to re-think what it means to count, as did your essay. Except the plant-fact is interesting while your essay is useful.
Plants do not count and have no awareness of time or of anything at all. The exact method by which venus fly traps activate is unknown but it seems hard to me to attribute it with the ability to count. That kind of teleological explanation is something we are cognitively biased to give but it fails to be explanatory.
Sunflowers do not turn their heads to face the sun because they want to catch more sunlight. They turn towards light because those cells that are in shadow receive more auxin which in turn stimulates the elongation of the cell walls causing the plant to grow in the opposite direction and towards the light. Natural selection will tend to favor those individuals that can gather more light than those which do not. There is no teleology involved.
Where are you seeing any teleology? Counting is just switching to a different state whenever a thing happens and performing a certain behavior when a certain state is reached. Time-sensitive behavior is… basically ubiquitous. (Yeah, yeah, “awareness” was a poor word choice.) You can buy counters and clocks at the electronics store! They don’t require any mysterious ghost of anthropomorphism!
Please allow me to clarify. “This plant moves as if it can count and is aware of time.” We are in agreement that the plant is not aware, and I was careful to say so. “Learning that caused me to re-think what it means to count, as did your essay. Except the plant-fact is interesting while your essay is useful.” Hearing a random song on the radio can spark a thought, but it doesn’t mean I’m thinking about that song. That’s what I mean by an interesting re-think. The essay I praise is something I’m thinking about. That’s what I mean by a useful re-think.
I like your first point.
I’m not sure what you’re saying with your second point. There are algorithms that can count the number of rocks on a beach (which take as input the graph of bonds between atoms. If you have an objection to counting bonds, we can instead talk about algorithms that count heaps of rocks based on the proximity of atoms). You could make any of the following objections to “three rocks”:
There is no canonical such algorithm, because such an algorithm needs a scale parameter. I would respond that there are ways for an algorithm to decide on an appropriate scale parameter.
There are many distinct such algorithms which agree when there are obviously three rocks, but disagree on where the boundary between “three rocks” and “two rocks” is. I would respond that this objection only applies to ambiguous arrangements of rocks.
There is a canonical family of rock-counting algorithms, but the fact that we care about rock-counting algorithms is a fact about ourselves, not a fact about the external world. I agree with this.
My second point is all about discretization. The ability to count the number of rocks comes after the ability to say “this is a rock”, and I argue that the reality does not come equipped with a “natural” criterion to form a distinguished set of atoms and proclaim it to be a rock. A discretization procedure must be supplied that will do this. The output of such a procedure is a mental construct that we perceive as a rock and call a rock; it is such constructs that we count.
I like your use of an algorithm to sharpen the question, but since it’s discretization that’s fundamental to my point, not the counting of objects, let’s consider an object recognition algorithm. These exist; supplied with an image of the beach with a rock on it, such an algorithm may be able to circle the rock accurately. We can imagine a really exact snapshot of information about all the atoms and their bonds at the beach (subject to the quantum mechanical constraints that aren’t the issue here and will be ignored), and a much more powerful algorithm of essentially the same kind that will process such a snapshot and delineate the rock for us.
I claim that such an algorithm comes with hardcoded constants that are specific to the human-style discretization procedure, and have no “natural” importance in reality. A scale parameter you mention is one such constant, but to my mind there are many more. For example, one way in which such an algorithm might determine the boundary of an object is by considering differences in matter density (air is much less dense than rock); but even though it may compute and compare the densities, there’s no natural way for it to decide which difference is significant and which is not. The choice of threshold will be appropriate to what our crude human senses are able to perceive, and what counts as significant in human experience. Or consider the inherently fuzzy boundary of the rock, at which the atomic behavior is very different from the interior of the rock (atoms leave in some quantities, air molecules seep in, perhaps there’s an oxide layer, perhaps some vapor’s forming...). When we perceive the rock as an object, we just don’t care about that; and the algorithm will have to decide in some manner: “this layer of atoms with very different behavior is small enough to be considered the boundary layer of the object rather than, say, analyzed as a completely different and separate object”. Well, the “small enough” is another threshold set at human values.
I disagree with you that “there are ways for an algorithm to decide on” appropriate values of all such constants/thresholds/parameters that does not utilize any human-specific values. I can see how this just might barely be possible with only the scale parameter, through the use of some ostensibly-objective constraints. E.g. a nano-creature faced with the rock will just “see” billions upon untold billions of separate “objects”, but, you might argue, if we require the algorithm to end up with a small number of recognized objects (i.e. just the one rock), it will be able to dial the scale parameter to roughly the human scale w/o hardcoding any human-specific numbers. But even that is something like a cop-out, because it’s not clear why we should expect to find a small number of objects rather than 10^10 of them; and in any case, I believe that if you properly account for the many separate parameters you need to tweak (or constants you need to set), this trick won’t work anyway.
You’re right!
I think a stronger point for your argument could be made by directly contrasting bringing things together and separating the world into things . Spatial separation is probably the first thing humans learn to do in order to count (and our vision apparently does this automatically for small numbers of things), and that can be followed by learning more abstract ways of discretizing. Spatially putting things next to each other to count them is valid so long as the separation remains to keep them distinct. Mentally moving our viewpoint is equivalent to moving the objects being counted next to each other. Glomming things back into a whole is what does not work, as you pointed out. Making it clear that the separation is what allowed the things to be counted in the first place seems like the most important thing to me.
Good work! I like your post more than Eliezer’s post that prompted it :-) If apples are mental constructs, next we need to figure out why we think they satisfy the second-order axiom of induction, and whether that reasoning can be automated.
“Subjective” and even “is in the map, not the territory” are tricky concepts, because they primarily invoke two connotations, one of which appears to be bogus and distorts the other: (1) the question is determined/disambiguated by some attribute of some agent, such as its preferences, emotions, perception, knowledge, prior, etc.; (2) the answer to the question is somewhat arbitrary, can be picked based on unrelated considerations, can’t be confidently declared incorrect, is “a matter of opinion”, etc. Once the question is clear enough, it doesn’t matter where the data that determined it came from, the question (or its meaning) screens off its origin.
The map-territory distinction can make this confusion more subtle by giving a clearer and thus more salient idea of the agent-associated data that would disambiguate a question. This data doesn’t make the clarified question a “part of the map”, even though the map may contain (a representation of) it, and it doesn’t make the map some sort of requisite for the question. The question is still about the world, we’ve just used a map to explain what the question is. But once we have the question, there is no need for that map anymore, which is the sense in which (I’m guessing that) I disagree with the following point from the post:
I think I mostly agree with your analysis and disagree that it contradicts what you quoted from mine. We both agree that the question is about the world, though its meaning comes from the map. At this point you say that “there is no need for that map anymore”, which I’m not sure I understand; I prefer to just note that we had to use the map to come up with the question. After all, my original goal was to show that the thought experiment “let’s imagine the Earth before there was life on it, and see, we have a strong intuition that these questions still have the same answers even then” is ill-defined: without maps, the questions can’t be asked.
(more on map/territory in this comment)
In this context, my point is that having a map in the future, or in a hypothetical future, or only in our world where we construct the question is enough to pose the question, and after that you can ask it about worlds that don’t have maps and would never have them. So the thought experiment won’t be ill-defined if you use (refer to) a human mind to pose it (and you do), even though that mind is not a part of the thought experiment.
Let me draw an analogy.
A: “Some say that judging the situation as funny or not funny is inextricably tied to humans and their particular values, but I think that’s not true. Imagine the Earth millions of years ago, before any humans were around, and imagine [insert a funny scene with some awkward behavior of dinosaurs and/or their offspring]. Even though no humans were around to laugh at these, I can’t help but think that the scene was funny. You don’t need humans to define what’s funny.”
B: “Some say that judging the arithmetical truth is inextricably tied to humans and their particular values, but I think that’s not true. [Repeat the scene with rocks on the beach billions of years ago]. I can’t help but think that 3 rocks + 2 rocks = 5 rocks in that scene. You don’t need humans to recognize 2+3=5 as valid”.
Now, your point works equally well on A’s and B’s stories. In A’s story, your point says that it’s possible to define the situation as funny (even though presumably we agree that funniness is very much anthropomorphic to a large degree) by reference to a map which is yet to come, or which exists outside of the imagined world. And I agree with that. But that’s not enough for A. What A wants to “sell” to you is the notion that funniness exists on its own terms within that imagined world, completely independent of the existence of humans and of their values. A really wants to see the situation as funny “on its own”, not because you, a human, is there/was there/will be there/could be there to judge it so.
Now, I happen to think that A’s position is indefensible, bordering on silly. Moreover, I imagine that B also feels that way about A’s statement. B thinks their statement is much more objective and “inherent” to the imagined world than A’s statement. B really wants to see 2+3=5 “objectively” embedded within the scene they’ve described without reference to hypothetical humans and their map to parse it, construct mental entities out of it, and count them. It won’t satisfy B if you say “Sure, there are 2+3=5 rocks on that beach, and that’s an objective statement about the situation, but only in the same sense as if you said the rocks were funny-shaped; in both cases you can say the questions are well-defined in Vladimir_Nesov’s sense, relying on the eventual/hypothetical humans coming along and providing a map to make sense of the questions”.
I think that B would disagree with that claim and consider their story much superior in “objectivity” to A’s. And my point is that B is wrong and A and B’s stories have the same status in that respect. I can present that status by saying the questions about funniness/number of rocks are ill-defined because no humans are around to parse them; or I can follow you and say they’re well-defined but only by virtue of humans eventually appearing to parse them. But I feel that there’s little to no difference between these two presentations; the important point is that B’s story has the same status as A’s.
And, since I’m apparently not satisfied with the length of this comment so far, I’ll reiterate that I don’t think this turns the laws of arithmetics into humanity-tied, “subjective” rules. I do think B’s final sentence is correct and you do not need humans to define 2+3=5 as valid. I just don’t see B’s story at all as evidence towards that conclusion. It doesn’t work as evidence for that. And that, I guess, is my original point 2.
It doesn’t feel right to call the spatial/counting distinction a “confusion”, as the spatial sense of counting is meaningful and doesn’t seem obviously unfit for its role in the discussion. It might be inappropriate for some reason, but that argument needs to be made. The post seems to mostly clarify the distinction, not motivate one option over the other.
Great post! I found this quite enlightening and easy to follow.
I pretty much agree with this analysis. I have one comment on 2 though.
Our minds have opinions on the following topics:
Whether or not something is a rock
Whether or not something is a valid candidate for the IsRock predicate
It’s worth pointing out that these are both somewhat arbitrary, with no right or wrong answer. If we say “yes this is a rock” or “no this doesn’t look like an IsRock predicate” then we’re expressing statements about our map/mind rather than about the territory itself.
That said, it feels like there is more than one predicate which:
You would feel is a valid IsRock candidate
Is talking about the territory rather than anyone’s map
Obeys counting axioms (subject to some complications such as already having a UnionOfTwoPortionsOfSpace predicate)
Just realised that I didn’t quite mean this.
If we say “this is/isn’t a valid IsRock predicate” then we’re expressing a statement about the map
Under one semantics, “this is a rock” means “I believe this is a rock” and is a statement about the map
By picking a predicate P which satisfies the IsIsRockCandidate metapredicate, we can give “this is a rock” the semantics P(This), which is a statement about the territory.
I like your attempt to see where the focus of our statements shifts back onto the territory, but in the crucial place I’m not sure how an IsRock predicate may or may not “obey counting axioms”—not sure what this means.
My take on this: a “rock” is a a concept that exists in the map, not in the territory. “Is a rock” (and, much more importantly and centrally, “Is an object”) is a predicate that exists within the map. However, “this is a rock” is a statement about the territory, not the map. How’s that work? When I state “this is a rock”, what I’m saying is that “out there” in reality there’s Stuff which, when mediated through my senses and subjected to the discretization process my map comes equipped with, will yield an Object that my map will recognize as a Rock. Not all possible configurations of Stuff in reality are such that this process will finish with a Rock in my map. By saying “this is a rock”, I claim a constraint on the Stuff in reality that ensures the successful completion of such a process (that I may or may not actually carry out).
Suppose the state of the universe at a particular time can be considered as a set of atoms in R^3.
Now let’s suppose we choose IsIsSomeRocksPredicate such that IsIsSomeRocksPredicate(P) implies:
P(r,u) is a binary predicate (the first argument being our candidate bunch of rocks and the second argument being the state of the universe)
Let T be an isometry. Then P(r,u) implies P(T(r), T(u))
P(r,u) implies r is a subset of u
P(r,u) implies that r is a finite set of atoms
Let u and u’ be identical in some neighborhood of r. Then P(r,u) implies P(r,u’)
P(r,u) implies that r is not solidly connected to anything within u (this stops you defining half a rock as a rock)
P(r,u) probably implies some other stuff about the essential rockiness of r.
Then we can pick a P that satisfies this, and come up with an IsSameNumberOfRocks equivalence function. We can also come up with a successor function (adding an extra rock) that is unique up to IsSameNumberOfRocks. A big problem is that the successor function isn’t always defined if there’s a finite number of rocks in the universe, so I guess rocks really don’t behave like numbers.
(I hadn’t realized just how far you need to unpack “2 rocks + 2 rocks = 4 rocks”… I’m interpreting it to mean something like “if you have 2 rocks, and then include another 2 rocks which don’t include any from the first set of 2, then you end up with the same number of rocks as if you picked an entirely different set of 4 rocks”
I’m not sure you’ve specified your definition fully.
Imagine there’s a biscuit that looks just like a rock. When filtered through your senses, you’ll recognize it as a rock and so by your definition it actually is a rock. Statements such as “this looks like a rock to me but actually it’s something else” become always false by definition.
(I’ll address the counting question separately)
I generally agree with point (1) but the point is irrelevant. Counting isn’t what makes 2 + 2 = 4 true. Although that is how we all learn to do math, by counting and memorizing addition and multiplication tables. I owe it all to my 3rd grade teacher. ;)
On point (2): “on our macro scale of reality, on the scale of things we perceive with our senses, discrete, separate objects are a feature of the map, not the territory; they exist in your mind, not the reality. In the reality, there’s just a lot of atoms everywhere”
There are no atoms at the macro scale. Or, if you like, atoms are everywhere. A chair is an “atom” of my dinning room furniture set and I can choose to count 5 items, four chairs and a table, or one item, one dinning room set. How I choose to cut up the world will determine which answer I get. But I am very confident that rocks and trees and universities and constitutions do not exist in my mind. They have an objective ontology that is independent of my personal subjective needs, interests and desires. Which is what it means for something to be real.
“Was 2+2=4 before humans were around to invent that equation?”
The statement: “2 + 2 = 4” is absolutely true because it is true in all possible worlds. Humans did not invent the equation, we invented the symbols and means of expressing it but the relation that is expressed in the words is an objective feature of the world that is true regardless of our opinions about it. Scientific facts have the world to word direction of fit. That is, they are true only to the extent they correspond to the world.
“we can certainly speak of single photons”
Only if we choose to observe them as particles. Photons have been observed experimentally to be both particles and waves. “The measurement apparatus detected strong nonlocality, which certified that the photon behaved simultaneously as a wave and a particle in our experiment. This represents a strong refutation of models in which the photon is either a wave or a particle.” This presents a significant challenge to certain theories.
I find myself able to agree with this in the case of rocks, but not in the case of apples. An apple doesn’t require the eye of the beholder. There’s no danger that a different observer, at a different scale, would claim that there are really two apples if you look close enough.
Apples do require categorization by an observer to some extent.
Is a nearly decayed apple still an apple? At what point does it stop being an apple? At what point does a fertilized apple blossom get to be called an apple?